ON UNIT SIGNATURES AND NARROW CLASS GROUPS OF ODD DEGREE ABELIAN NUMBER FIELDS
BENJAMIN BREEN, ILA VARMA, AND JOHN VOIGHT (APPENDIX WITH NOAM ELKIES)
Abstract. For an abelian number field of odd degree, we study the structure of its 2- Selmer group as a bilinear space and as a Galois module. We prove structural results and make predictions for the distribution of unit signature ranks and narrow class groups in families where the degree and Galois group are fixed.
Contents
1. Introduction 1
2. Properties of the 2-Selmer group and its signature spaces 8
3. Galois module structures 11
4. Galois module structures for odd degree extensions 14
5. Galois module structures for odd degree abelian extensions 18
6. Conjectures 26
7. Computations 31
Appendix A. Cyclic cubic fields with signature rank 1 (with Noam Elkies) 34
References 40
1. Introduction
1.1. Motivation. Originating in the study of solutions to the negative Pell equation, the investigation of signatures of units in number rings dates back at least to Lagrange. While a considerable amount of progress has been made for quadratic fields [38,21,7], predictions for the distribution of narrow class groups and possible signs of units under real embeddings for certain families of higher degree number fields have only recently been developed [16, 14,4].
In this paper, we study unit signatures and class groups of abelian number fields of odd degree. To illustrate and motivate our results, we begin with two special cases.
Conjecture 1.1.1 (Conjecture 6.3.3). As K varies over cyclic cubic number fields, the probability that K has a totally positive system of fundamental units is approximately 3%.
For the conjectures presented in this paper, we sidestep the issue of ordering fields: we expect that anyfair counting function in the terminology of Wood [41] should be allowed, for example ordering by conductor or by the norm of the product of ramified primes. We are led to Conjecture 1.1.1 by combining structural results established herein with a randomness
Date: April 8, 2021.
2020Mathematics Subject Classification. 11R29, 11R27, 11R45, 11Y40.
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hypothesis (H2) in the vein of the Cohen–Lenstra heuristics. This conjecture agrees well with computational evidence (see section 7.1); the following theorem provides additional theoretical support.
Theorem 1.1.2 (TheoremA.1.2, with Elkies). There exist infinitely many cyclic cubic fields with a totally positive system of fundamental units.
The proof of Theorem 1.1.2 involves the study the integral points on a log K3 surface.
The (infinite) family of simplest cubic fields of Shanks were each shown to have units of all possible signatures by Washington [40, p. 371], the case complementary to Theorem 1.1.2.
Our second illustrative conjecture is as follows.
Conjecture 1.1.3 (Conjecture 6.1.1). As K ranges over cyclic number fields of degree 7 with odd class number, the probability that the narrow class number is also odd is 7/9.
We recall that the narrow class group of a field coincides with the class group if and only if there are units of all possible signatures. This conjecture also matches computational evidence well (see section 7.2).
The predictions above are based on the philosophy underlying the Cohen–Lenstra heuris- tics, which predicts random behavior for arithmetic objects as soon as one accounts for all of the determined structure. Early examples of the need to account for structural proper- ties, including genus theory and ranks of units, were already present in the original paper of Cohen–Lenstra [9]. It remains mysterious and important to understand how one must account for additional structure in generalizations of the Cohen–Lenstra heuristics. For ex- ample, what makes a primegood [11], and the interaction betweenp-parts of class groups and the presence of pth roots of unity [32, 33, 2], remain unresolved. On the other hand, some reflection principles like those of Scholz and Leopoldt [28], seem to be inherently compatible with the Cohen–Lenstra–Martinet conjectures [17, 26, 27].
In this paper, we propose a model for the distribution of 2-parts of narrow class groups and signatures of units in families of abelian number fields of fixed odd degree and Galois group. For such families, the Galois action and the presence of the 2nd roots of unity suggest additional, nontrivial structure to account for (as confirmed by computations and available function field analogues). The requirement that the degree is odd when p = 2 isolates the
“roots of unity problem” from other obstructions to arithmetic randomness, including genus theory. Since the narrow class group is an extension of the class group by an elementary abelian 2-group that measures signatures of units, our efforts are concentrated on 2-parts.
Our contributions are thus twofold. First, for these families we precisely identify and analyze relevant structure, including the relationship to reflection principles. Second, under the hypothesis that what remains behaves randomly, we make exact predictions for the behavior of units and class groups—with corroborating computational evidence.
1.2. Structure: class groups. We now set up the structures we study and model in this paper. We build on work of Dummit–Voight [16], who make predictions for fields of odd de- greenwhose Galois closure has Galois groupSn. Here, we instead consider Galois extensions of odd degree.
Recall that the 2-Selmer groupof a number field K is
Sel2(K) :={α∈K×: (α) = a2 for a fractional ideala of K}/(K×)2.
Attached to K is a finite-dimensional F2-vector space V∞(K) V2(K) equipped with a nondegenerate symmetric bilinear form and a homomorphism
ϕK: Sel2(K)→V∞(K)V2(K)
called the 2-Selmer signature map. Dummit–Voight [16] showed that the image S(K) :=
img(ϕK) of ϕK is a maximal, totally isotropic subspace. If K is Galois over Q with Galois groupGK, then we observe that the above objects carry aGK-action and in particularS(K) is a GK-invariant, maximal totally isotropic subspace (Corollary 3.1.3).
In preparation for stating our guiding result, we introduce a bit of notation. Let G be a finite abelian group of odd order and let χ be an F2-character of G. Then every irreducible F2[G]-module is isomorphic to F2(χ), the value field of an F2-character χ: G → F
×
2 taking values in a (fixed) algebraic closure F2 of F2, where G acts through the character χ. For a finitely generatedZ[G]-moduleM, write rkχM for the multiplicity of the irreducible module F2(χ) in the F2[G]-module M/M2, and let rk2M := dimF2M/M2. For an F2-character χ of G, there is a noncanonical F2[G]-module isomorphism HomF2(F2(χ),F2) ' F2(χ−1) (see Lemma5.1.2 and the discussion preceding it), and we writeχ∗ :=χ−1 for the corresponding dual character. We say χ is self-dual if F2(χ∗) ' F2(χ) as F2[G]-modules. For an F2- characterχofGandV anF2[G]-module, we writeVχfor the F2(χ)-isotypic component ofV andVχ± :=Vχ+Vχ∗ for the sum. IfV is equipped with a symmetric,G-invariantF2-bilinear form, then the decomposition of V into the spaces Vχ± is orthogonal (Lemma 3.2.7), giving a canonical decomposition as F2[G]-modules.
Now let K be a Galois number field with abelian Galois groupGK of odd order; then the class group Cl(K) and narrow class group Cl+(K) are Z[GK]-modules. For anF2-character χ, we define the following nonnegative integers:
ρχ(K) := rkχCl(K);
ρ+χ(K) := rkχCl+(K);
kχ+(K) := rkχCl+(K)−rkχCl(K)
=ρ+χ(K)−ρχ(K).
(1.2.1)
We refer tok+χ(K) as theχ-isotropy rank. WhenK is clear from context, we drop it from the notation. Our main theorem, governing the above structures and quantities, is as follows.
Theorem 1.2.2 (Theorem 5.4.2). Let K be a Galois number field with abelian Galois group GK of odd order. Then for eachF2-characterχ, there are exactly6possibilities forS(K)χ± ≤ V∞(K)V2(K) up to GK-equivariant isometry.
Theorem 1.2.2 follows from an investigation of the F2[GK]-module structure of the 2- Selmer signature map together with a classification of invariant, maximal isotropic subspaces in a bilinear space with group action. The six possibilities are given in Table 1.2.3: we write q := #F2(χ) = #F2(χ∗), and we write # IsomG(V) for the number of G-equivariant isometries ofV(K) :=V∞(K)V2(K). All cases occur (see Example6.4.1), so the statement is optimal in this sense.
In Tables 1.2.3 and 1.2.4 we observe parallel relations when V∞(K) is replaced by V2(K) and Cl+(K) is replaced by Cl4(K), the ray class group of K of conductor 4, with the quantities ρ4,χ(K) and k4,χ(K) defined analogously as in (1.2.1); we restrict attention to narrow class groups in this introduction.
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Case χ self-dual? Sχ± Sχ±∩V∞ Sχ±∩V2 # IsomG(V)
A Yes F2(χ) {0} {0} √
q+ 1
B No F2(χ)2 F2(χ) F2(χ) 1
B0 No F2(χ∗)2 F2(χ∗) F2(χ∗) 1 C No F2(χ)⊕F2(χ∗) F2(χ) F2(χ∗) 1 C0 No F2(χ)⊕F2(χ∗) F2(χ∗) F2(χ) 1
D No F2(χ)⊕F2(χ∗) {0} {0} q−1
Table 1.2.3: Possibilities for the Galois bilinear structure of the image of the 2-Selmer group Case ρχ and ρχ∗ ρ+χ and ρ+χ∗ ρ4,χ and ρ4,χ∗ kχ+ kχ+∗ k4,χ k4,χ∗
A ρχ=ρχ∗ ρ+χ =ρ+χ∗ ρ4,χ =ρ4,χ∗ 0 0 0 0 B ρχ =ρχ∗+ 1 ρ+χ =ρ+χ∗ ρ4,χ =ρ4,χ∗ 0 1 0 1 B0 ρχ =ρχ∗ −1 ρ+χ =ρ+χ∗ ρ4,χ =ρ4,χ∗ 1 0 1 0 C ρχ=ρχ∗ ρ+χ =ρ+χ∗−1 ρ4,χ =ρ4,χ∗+ 1 0 1 1 0 C0 ρχ=ρχ∗ ρ+χ =ρ+χ∗ + 1 ρ4,χ =ρ4,χ∗−1 1 0 0 1 D ρχ=ρχ∗ ρ+χ =ρ+χ∗ ρ4,χ =ρ4,χ∗ 0 0 0 0
Table 1.2.4: Possibilities for the class group and isotropy rank The following corollary is then immediate.
Corollary 1.2.5. Under the hypotheses of Theorem 1.2.2, we have
|rkχCl(K)−rkχ∗Cl(K)| ≤1,
|rkχCl+(K)−rkχ∗Cl+(K)| ≤1, and 0≤k+χ +k+χ∗ ≤1.
Corollary 1.2.5 can be seen as a Spiegelungssatz or reflection theorem as in Leopoldt [28]
forp= 2, and therefore Theorem 1.2.2can be seen as a precise refinement of it. A precursor to Corollary 1.2.5 is the theorem of Armitage–Fr¨ohlich [1], generalized by Taylor [39] and Oriat [34, 35]. Gras then proved a very general T-S-reflection principle [23, Th´eor`eme 5.18]
(see also the presentation in his book [24, Chapter II, Theorem 5.4.5]); however, certain corollaries for p = 2 [24, Chapter II, Corollary 5.4.6(ii)] (details [24, Chapter II, (5.4.9)]
added in the second printing) are incorrect: case D of Table 1.2.3 does not appear.
We show that rank inequalities like Corollary 1.2.5 for a Galois number field K of odd degree follow from Kummer duality and the GK-module structure of the 2-Selmer group (and its intersection with coordinate subspaces in the 2-Selmer signature space). In partic- ular, the relevant reflection principles are already encoded. In particular, we recover easily several classical results from the literature. Our results can also instead be seen to fit into a much more general context (Poitou–Tate duality of Selmer groups); however, in view of the subtleties indicated in the previous paragraph, one advantage of our approach is it provides a self-contained, uniform, and transparent proof of these corollaries. At the same time, the concrete description in Theorem 1.2.2 states the precise structure (in particular, the image of the 2-Selmer group under the signature map is a GK-invariant, maximal totally isotropic
subspace) which must be respected in a random model and thereby serves as the foundation for our heuristics, which we present in sections 1.4–1.5.
1.3. Structure: units. The structural result in Theorem 1.2.2 has the following conse- quence for units. Let OK be the ring of integers of K. The archimedean signature map sgn∞: K× →Q
v|∞{±1} 'Fn2 is the surjective group homomorphism recording the signs of elements of K× under each real embedding; its kernel K>0× := ker(sgn∞) is the group of totally positive elements of K. Let OK,>0× :=O×K∩K>0× denote the group of totally positive units. Define
sgnrkχ(O×K) := rkχsgn∞(OK×), (1.3.1) and theunit signature rank of K
sgnrk(OK×) := dimF2sgn∞(OK×) =P
χsgnrkχ(OK×)·[F2(χ) :F2], (1.3.2) where the sum indexes over isomorphism classes of F2-characters χ. The structure on unit signature ranks imposed by the Galois module structure is summarized in the following result, keeping the notation (1.2.1).
Theorem 1.3.3 (Theorem5.5.2). LetK be an abelian number field of odd degree with Galois group GK, and let χ be an F2-character of GK. Then the following statements hold.
(a) If kχ+(K) = 1, then sgnrkχ(O×K) = 0.
(b) If kχ+(K) = 0, then 1−rkχCl(K)≤sgnrkχ(OK×)≤1.
When the degree of K is prime, summing over χ gives the following corollary.
Corollary 1.3.4 (Corollary 5.5.4). Let K be a cyclic number field of odd prime degree `, and let f be the order of 2 modulo `. Then
sgnrk(OK×)≡1 (mod f), and the following statements hold.
(a) If f is odd, then `+12 −rk2Cl(K)≤sgnrk(OK×)≤`;
(b) If f is even, then `−rk2Cl(K)≤sgnrk(O×K)≤`.
For example, if 2 is a primitive root modulo ` and the class number of K is odd, then sgnrk(O×K) =`; this result for` = 3 was observed by Armitage–Fr¨ohlich [1, Theorem V].
1.4. Heuristics: narrow class groups. We begin by applying the results in the previous section to make predictions for narrow class groups and signatures of units for odd-degree abelian number fields. We keep the notation of (1.2.1).
Let G be a finite abelian group of odd order. A G-number field is a Galois number field K, inside a fixed algebraic closure of Q, equipped with an isomorphism GK ' G, where GK := Gal(K|Q). Such a field K is totally real, so ±1 are the only roots of unity in K.
Returning to Theorem1.2.2and Table1.2.3, we see that the quantitieskχ+, k+χ∗are uniquely determined bySχ± in the cases whereχis self-dual or casesBand B0 whenχis not self-dual.
However, when χ is not self-dual and ρχ = ρχ∗, there is a question about the distribution of cases C, C0, and D. Modeling the image of 2-Selmer signature map as a random totally isotropic G-invariant subspace in the 2-Selmer signature space (see heuristic assumption (H1)), we are led to the following conjecture.
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Conjecture 1.4.1 (Conjecture 6.1.1). Let G be an abelian group of odd order, and let χ be an F2-character of G that is not self-dual and let q := #F2(χ). Then as K varies over G-number fields satisfying ρχ(K) = ρχ∗(K),
Prob kχ+(K) +k+χ∗(K) = 0
= q−1 q+ 1; Prob kχ+(K) +k+χ∗(K) = 1
= 2
q+ 1.
(1.4.2)
A concrete application of Conjecture1.4.1is given in Conjecture6.1.2, as follows. Suppose 2 has order (`−1)/2 modulo a prime` ≡7 (mod 8): then there are exactly two non-self-dual characters, and if Cl(K) is self-dual then kχ = kχ∗ = 0. So as K varies over cyclic number fields of degree ` such that Cl(K)[2] is self-dual, Conjecture 1.4.1 predicts that
Prob Cl+(K)[2]'Cl(K)[2]) = 2`−12 −1
2`−12 + 1. (1.4.3)
We further expect that the probability in Conjecture 1.4.1 remains the same in certain natural subfamilies, such as when we fix the value rkχCl(K) = rkχ∗Cl(K) =r. As a special case, we arrive at Conjecture1.1.3.
1.5. Heuristics: units. Next, we make predictions for signatures of units. Our model can be applied under many scenarios; in this introduction, we consider two simple, illustrative cases. We first examine the situation when the degree is prime and the class number is odd. Modeling O×K/(O×K)2 as a random GK-invariant subspace of the 2-Selmer group of K containing −1, and under an independence hypothesis (H20), we are led to the following conjecture.
Conjecture 1.5.1 (Conjecture 6.2.4). Let ` be an odd prime such that the order f of 2 in (Z/`Z)× is odd. Let q := 2f, and define m := `−12f ∈ Z>0. Then as K varies over cyclic number fields of degree ` with odd class number,
Prob sgnrk(OK×) = f s+`−12
= m
s
q−1 q+ 1
s 2 q+ 1
m−s
for 0≤s≤m.
Second, we consider the situation when ` = 3 or 5 with no additional assumption on the class number. In this case, Corollary 5.5.4(b) implies that sgnrk(O×K) = 1 or `. Although complete heuristics for the 2-part of the class group over abelian fields are not known, Malle [32] provides results in the case that ` = 3 or 5. We use the following notation: for m∈Z≥0∪{∞}andq∈R>1, write (q)0 := 1 and otherwise (q)m :=Qm
i=1(1−q−i). Combining these results with a uniform random hypothesis (H2), we make the following prediction.
Conjecture 1.5.2 (Conjecture 6.3.3). Let `= 3 or 5 andq = 2`−1. As K varies over cyclic number fields of degree `, then
Prob sgnrk(O×K) = 1
=
1 + 1
√q (√
q)∞(q2)∞
(q)2∞ ·
∞
X
r=0
(`−1)r q(r2+3r)/2 ·(q)r
· qr−1
qr+1−1. (1.5.3)
Computing the numerical value of the quantity in (1.5.3), we predict that approximately 3% of cyclic cubic fields have sgnrk(O×K) = 1 which yields Conjecture 1.1.1. For cyclic quintic fields we predict that this proportion drops to below 0.1%. The predictions in the two conjectures above agree with the computational evidence we compiled: see section 7.
Remark 1.5.4. To extend the above conjectures to all odd primes (or more generally, to all abelian groups G of odd order), we would need to refine the heuristics of Malle [32, 33] to predict the distribution of rkχCl(K). This distribution will depend on the representation theory ofZ/`Z(or more generally, ofG); in particular, the constraints in Theorem1.2.2must be respected. In contrast, when 2 is a primitive root modulo `, there is only one nontrivial (necessarily self-dual) F2[Z/`Z]-module, so these representation-theoretic complexities are immaterial; in this case, we expect that the generalization of the above conjectures to such
` to be more straightforward.
We expect that as ` → ∞ varies over odd primes, we have Prob sgnrk(OK×) = 1
→ 0%, and we plan to give evidence to support this limiting behavior in the future (see also Remark 7.2.3).
Remark 1.5.5. The statements we prove and conjecture above on unit signature ranks in odd degree extensions are quite different than the situation for real quadratic fields, related to solutions to the negative Pell equation. By genus theory, 100% of real quadratic fields have a totally positive unit [21], and the conjectural asymptotic due to Stevenhagen [38] arises from an apparently different heuristic involving R´edei matrices.
Remark 1.5.6. We are not aware of a function field analogue which would bear on the conjectures presented in this section. These conjectures are based on structural properties of the 2-Selmer signature map, which rely in an essential way on the fact that 2 ∈ OK is neither a unit nor zero.
1.6. Outline. In section2, we set up basic notation and background. In section3, we study these objects in general as Galois modules overF2. We then restrict to the case of odd Galois extensions in section 4and show how reflection principles follow from the Galois action and Kummer duality—these are for completeness (and to indicate that they are not missing from our model). We then further restrict to abelian extensions and in section 5 prove our main structural result, and we see classical reflection principles as a corollary. In section 6 we introduce our heuristic assumptions and present our conjectures, including details on the low-degree cases. In section 7, we carry out computations that provide some experimental evidence for our conjectures. Finally, in appendix A we prove Theorem 1.1.2.
1.7. Acknowledgements. The authors would like to thank Edgar Costa, David Dummit, Noam Elkies, Georges Gras, Brendan Hassett, Hershy Kisilevsky, Evan O’Dorney, Arul Shankar, Jared Weinstein, and Melanie Matchett Wood for comments, and Tommy Hofmann for sharing his list of cyclic septic fields. Special thanks go to two anonymous referees for their excellent and detailed feedback and suggestions. Varma was partially supported by an NSF MSPRF Grant (DMS-1502834) and an NSF Grant (DMS-1844206). Voight was supported by an NSF CAREER Award (DMS-1151047) and a Simons Collaboration Grant (550029). Elkies was partially supported by an NSF grant (DMS-1502161) and a Simons Collaboration Grant.
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2. Properties of the 2-Selmer group and its signature spaces
We begin by setting up some notation and recalling basic definitions and previous results.
2.1. Basic notation. If A is a (multiplicatively written) abelian group and m ∈ Z>0, we write
A[m] :={a∈A:am = 1}
for the m-torsion subgroup of A. For a prime p, we write rkp(A) := dimFp(A/Ap) for the p-rankof A; we then have #A[p] =prkp(A).
We quickly prove a standard lemma (for lack of a reference). LetZ(p) :={a/b∈Q:p-b}
be the localization of Z away from a prime p.
Lemma 2.1.1. LetGbe a finite group, letp-#Gbe prime, and letM be a finitely generated, torsion Z(p)[G]-module. Then there is a (noncanonical) isomorphism M/pM −→∼ M[p] as Fp[G]-modules.
Proof. Recall (by Maschke’s theorem) that every finitely generatedFp[G]-module is semisim- ple, since p-#G. Let m=pr be the exponent ofM (as an abelian group), with r≥0. We argue by induction on r. If r ≤1, thenpM ={0} so indeed M/pM =M =M[p].
Suppose the result holds whenever M has exponent dividing pr; we prove it for M of exponent pr+1. Multiplication by p gives an exact sequence
0→M[p]→M →pM →0
of Z(p)[G]-modules. We can repeat this with pM, giving the following diagram, with exact rows and columns:
0 0 0
0 (pM)[p] pM p2M 0
0 M[p] M pM 0
0 M1 M/pM pM/p2M 0
0 0 0
(2.1.2)
Here, M1 := coker((pM)[p] → M[p]) ' M[p]/(pM)[p]. By semisimplicity, the left vertical and bottom horizontal maps split, so
M[p]'(pM)[p]⊕M1
M/pM '(pM/p2M)⊕M1 (2.1.3)
Since pM has exponent pr, by induction (pM)[p]'(pM/p2M) so M[p]'M/pM.
Let K be a number field of degree n = [K : Q] with r1 real and r2 complex places, with algebraic closureK and with ring of integers OK. For a prime p, we denote the localization of OK away from (p) by
OK,(p):={α∈K× : ordp(α)≥0 for all primesp|(p)}
and the completion of OK at p by OK,p :=OK⊗Zp, so that OK,(p) ,→ OK,p. For a place v of K, we let Kv denote the completion of K atv and OK,v its valuation ring, and we let
( , )v: Kv××Kv× → {±1}
denote the Hilbert symbol atv: recall that forαv andβv ∈Kv×, we have (αv, βv)v = 1 if and only if βv is in the image of the norm map fromKv[x]/(x2−αv) to Kv.
2.2. The 2-Selmer group and its signature spaces. The main object of study is the 2-Selmer groupof a number field K, defined as
Sel2(K) :={α∈K×: (α) = a2 for a fractional ideala of K}/(K×)2.
Following Dummit-Voight [16, Section 3], we recall two signature spaces that keep track of behavior at ∞ and at 2, as follows.
Definition 2.2.1. The archimedean signature space V∞(K) is defined as V∞(K) := Y
vreal
{±1} 'Y
v|∞
Kv×/Kv×2
where the second product runs over all real places ofK. Thearchimedean signature map is sgn∞: K× →V∞(K)
α 7→(sgnv(α))v where sgnx=x/|x| for x∈R×.
By definition, ker sgn∞ = K>0×, the totally positive elements of K, which contains (K×)2, and so the map sgn∞ induces a well-defined map ϕK,∞: Sel2(K)→V∞(K). The product of Hilbert symbols defines a map
b∞: V∞(K)×V∞(K)→ {±1}
which is a (well-defined) symmetric, non-degenerate F2-bilinear form.
Definition 2.2.2. The 2-adic signature spaceV2(K) is defined as V2(K) := OK,(2)× /(1 + 4OK,(2))(O×K,(2))2. The 2-adic signature map is the map
sgn2: O×K,(2) →V2(K) obtained from the projection O×K,(2) →V2(K).
For the following statements we refer to Dummit–Voight [16,§4]. We have dimF2V2(K) = n and there is an isomorphism of abelian groups
V2(K)' Y
v|(2)
O×K,v/(1 + 4OK,v)(OK,v× )2.
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Under this identification, the product of Hilbert symbols defines a map b2: V2(K)×V2(K)→ {±1}
(αv)v,(βv)v
7→ Y
v|(2)
(αv, βv)v,
which is a nondegenerate, symmetric F2-bilinear form on V2(K). Every class in Sel2(K) has a representative α such that α ∈ O×K,(2), unique up to multiplication by an element of (OK,(2)× )2; therefore, the map sgn2 induces a well-defined map ϕK,2: Sel2(K)→V2(K).
Putting these together, we define the2-Selmer signature spaceas the orthogonal direct sum V(K) :=V∞(K)V2(K)
and write b := b∞ ⊥ b2 for the bilinear form on V(K). The isometry group of (V(K), b) is the product of the isometry groups (or equivalently, the subgroup of the total isometry group preserving each factor). Equipped with b, the 2-Selmer signature space V(K) is a nondegenerate symmetric bilinear space over F2 of dimension r1+n. Similarly, we define the 2-Selmer signature map
ϕK :=ϕK,∞⊥ϕK,2: Sel2(K)→V(K). (2.2.3) Theorem 2.2.4 (Dummit–Voight [16, Theorem 6.1]). For a number field K, the image of the 2-Selmer signature map ϕK is a maximal totally isotropic subspace.
Recall from the introduction that the class group of K is denoted by Cl(K), its narrow class group is denoted by Cl+(K), and its ray class group of conductor 4 by Cl4(K).
Definition 2.2.5. The archimedean isotropy rankof a number field K is k+(K) := rk2Cl+(K)−rk2Cl(K),
and the 2-adic isotropy rank of K is
k4(K) := rk2Cl4(K)−rk2Cl(K).
By Dummit–Voight [16, Theorem 6.1], we have
k+(K) = dimF2img(ϕK)∩V∞= dimF2ker(ϕK,2)−dimF2ker(ϕK) k4(K) = dimF2img(ϕK)∩V2 = dimF2ker(ϕK,∞)−dimF2ker(ϕK)
hence the nomenclature given in Definition 2.2.5. Moreover, there is a classical equality
k4(K) =k+(K) +r2 (2.2.6)
(see for example, Theorem 2.2 of Lemmermeyer [29] and also Theorem 4.3.3 below).
2.3. Connections to Sel2(K) via class field theory. There is a natural, well-defined map Sel2(K) → Cl(K)[2] sending [α] 7→ [a] where a2 = (α); this map is surjective and fits into the exact sequence
1→ O×K/(OK×)2 →Sel2(K)→Cl(K)[2]→1. (2.3.1) In addition, the 2-Selmer signature map arises naturally in class field theory as follows.
Let H ⊇ K be the Hilbert class field of K. Class field theory provides an isomorphism
Gal(H|K) 'Cl(K); let H(2) denote the fixed field of the subgroup Cl(K)2. The Kummer pairing
Gal(H(2)|K)×ker(ϕK)→ {±1}
(τ,[α])7→ τ(√
√α) α
is (well-defined and) perfect [16, (3.11)]. The Artin reciprocity map provides a canonical isomorphism Gal(H(2)|K) ' Cl(K)/Cl(K)2 and so we can rewrite the above map instead as
Cl(K)/Cl(K)2×ker(ϕK)→ {±1}. (2.3.2) The pairing (2.3.2) is the first of four perfect pairings [16, Lemma 3.10] (see also Lemmer- meyer [29, Theorem 6.3]); the other three perfect pairings are
Cl4(K)/Cl4(K)2×ker(ϕK,∞)→ {±1}, Cl+(K)/Cl+(K)2×ker(ϕK,2)→ {±1}, and
Cl+4(K)/Cl+4(K)2×Sel2(K)→ {±1},
(2.3.3)
where Cl+4(K) denotes the ray class group of K of conductor 4· ∞.
3. Galois module structures
We next study the Galois module structure on the arithmetic objects introduced in the previous section; we will continue in the next section with more precise results in the odd degree case. Our results overlap substantially with those of Taylor [39].
From now on, suppose thatK is Galois overQ, with Galois group GK := Gal(K|Q). We work throughout with left F2[GK]-modules. (We could consider more generally structures implied by the action of a nontrivial automorphism group Aut(K), and many of the results below could be generalized to this setting; we focus here on the extreme case, where Aut(K) is as large as possible.)
3.1. Basic invariants. We first prove Galois invariance of the signature spaces in generality.
Recall that aF2-bilinear formb:V ×V →F2 on anF2[G]-moduleV isG-invariantifb(α, β) = b(σ(α), σ(β)) for all σ∈G.
Proposition 3.1.1. The following statements hold.
(a) If K is totally real, then V∞(K) 'F2[GK] as F2[GK]-modules; otherwise, V∞(K) is trivial. In either case, the bilinear form b∞ is GK-invariant.
(b) We have V2(K)'F2[GK] as F2[GK]-modules, and b2 is GK-invariant.
Proof. We begin with (a), and suppose that K is totally real. The Galois group GK acts onV∞(K) (on the left) via its permutation action on the (index) set of real places ofK (as v 7→ v ◦σ−1), so V∞(K) ' F2[GK] as F2[GK]-modules. Since b∞ is defined as the product over real placesv, it is GK-invariant.
For (b), we follow the proof in Dummit–Voight [16, Proposition 4.4]. The map a7→1 + 2a induces an isomorphism
OK,(2)/2OK,(2) −→ O∼ ×K,(2)/(1 + 4OK,(2)) [2]
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which is visiblyGK-equivariant. By Lemma2.1.1, the right hand side is isomorphic toV2(K) as F2[GK]-modules, since (1 + 4OK,(2))2 ≤1 + 8OK,(2) ≤ O×2K,(2).
Finally, we show b2 is GK-invariant. Let α, β ∈ O×K,(2) and let v | 2 be a prime of K.
Since GK acts transitively (on the left) on the set of places {v : v | (2)} with stabilizers Dv := Aut(Kv) the decomposition group, choosing a place v we have
b2(α, β) = Y
τ Dv∈GK/Dv
(α, β)τ(v)
well defined. The Hilbert symbol ( , )v is GK-equivariant and Dv-invariant, so for σ∈GK, b2(σ(α), σ(β)) = Y
τ Dv∈GK/Dv
(σ(α), σ(β))τ(v) =Y
τ
(α, β)(σ−1τ)(v) =Y
τ
(α, β)τ(v)=b2(α, β)
since σ permutes the cosets of Dv inGK.
Lemma 3.1.2. The 2-Selmer signature map ϕK isGK-equivariant.
Proof. We show that both sgn∞ and sgn2 are G-equivariant which implies that the induced maps ϕK,∞ and ϕK,2 are G-equivariant as well. For sgn∞, we may suppose thatK is totally real, and then
sgn∞(σ(α)) = (sgn(v(σ(α))))v = (sgn((σ−1v)(α)))v =σ(sgn∞(α)) so sgn∞ is GK-equivariant.
To show that sgn2 isGK-equivariant, we observe that sgn2 is simply the composition of a
natural embedding and projection.
Corollary 3.1.3. For a Galois number field K, the image of the 2-Selmer signature map ϕK is a GK-invariant maximal totally isotropic subspace.
Proof. Combine Theorem2.2.4 with Proposition 3.1.1 and Lemma 3.1.2.
3.2. Duals and pairings. We now treat some issues of duality, with an application to the Kummer pairing. Let G be a finite group and let V be a finitely generated (left) F2[G]- module.
Definition 3.2.1. The dualof V is the F2-vector space V∨ := HomF2(V,F2)
equipped with the (left) F2[G]-action, arising from extending F2-linearly the natural G- action: if σ ∈ G, f ∈ V∨, and x ∈ V, then (σf)(x) := f(σ−1x). We say V is self-dual if V 'V∨ as F2[G]-modules.
The canonical evaluation pairing
e: V∨×V →F2
e(f, x) = f(x) (3.2.2)
is nondegenerate and G-invariant, so gives a canonical isomorphism V −→∼ (V∨)∨ as F2[G]- modules.
Lemma 3.2.3. For K a Galois number field, the Kummer pairings (2.3.2)–(2.3.3) induce canonical isomorphisms of F2[GK]-modules:
Cl(K)/Cl(K)2 ' ker(ϕK)∨ Cl4(K)/Cl4(K)2 ' ker(ϕK,∞)∨ Cl+(K)/Cl+(K)2 ' ker(ϕK,2)∨ Cl+4(K)/Cl+4(K)2 ' Sel2(K)∨
(3.2.4)
Proof. We work with the first line, the others follow by the same argument. The Kummer isomorphism
K×/K×2 −→∼ Hom(Gal(K|K),{±1}) (3.2.5) is GK-equivariant and defines a canonical isomorphism ker(ϕK)−→∼ Hom(Gal(Q|F),{±1}), where Qis the maximal subfield whose Galois group has exponent dividing 2 in the Hilbert class field ofK. The Artin map defines a canonicalGK-equivariant isomorphism Gal(Q|F)−→∼ Cl(K)/Cl(K)2. Combining these with the evaluation map then gives a canonical pairing
Cl(K)/Cl(K)2×ker(ϕK)→ {±1} (3.2.6) as claimed. This pairing may be explicitly described as
([a],[α])7→α a
where a ⊆ OK is an ideal of odd norm, α ∈ OK is coprime to a, and α a
is the Jacobi
symbol.
Applying Lemma2.1.1to the groups on the left-hand side of (3.2.4) gives (now noncanon- ical) F2[GK]-module isomorphisms Cl(K)[2]'ker(ϕK)∨, etc.
Lemma 3.2.7. Let b: V ×V → F2 be a G-invariant F2-bilinear form, and let W, W0 ⊆ V be irreducible F2[G]-modules. If W∨ 6'W0 as F2[G]-modules, then b(W, W0) ={0}.
Proof. Restrictingb, we obtain anF2[G]-module mapW0 →W∨ byw0 7→b( , w0); by Schur’s lemma, this map is either zero or an isomorphism, and the result follows.
Lemma 3.2.7, although easy to prove, is fundamental in what follows: it shows that when a decomposition of V into irreducibles is possible, it is already almost an orthogonal decomposition.
To conclude this section, we refine this into a canonical orthogonal decomposition. Suppose G has odd order, so the category of F2[G]-modules is semisimple. Let W be an irreducible F2[G]-module. We write VW for the W-isotypic component of V in a decomposition of V into irreducibles. Suppose that V is equipped with a symmetric, G-invariant, F2-bilinear form. Then by Lemma 3.2.7 we have a canonical decomposition as F2[G]-modules
V '
W (VW +VW∨), (3.2.8)where the orthogonal direct sum is indexed by irreduciblesW up to isomorphism and duals.
We call the decomposition given in (3.2.8) the canonical orthogonal decomposition of V.
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4. Galois module structures for odd degree extensions
In this section, we suppose throughout thatK hasodd degree (but remains Galois). Then K is totally real and the only roots of unity in K are ±1. Moreover, since #GK is odd, the category of leftF2[GK]-modules is semisimple.
4.1. Basic invariants. We quickly prove two standard lemmas, for completeness.
Lemma 4.1.1. We have O×K/(O×K)2 'F2[GK] as F2[GK]-modules.
Proof. We considerO×K as a Z[GK]-module multiplicatively. By Dirichlet’s unit theorem, (O×K/{±1} ⊗ZR)⊕R'R[GK]
asR[GK]-modules whereRhas trivialGK action (corresponding to the trace zero hyperplane in the Minkowski embedding). Counting idempotents, we conclude that
(O×K/{±1} ⊗ZZ(2))⊕Z(2) 'Z(2)[GK] (4.1.2) as Z(2)-modules; tensoring (4.1.2) with Z/2Zand using that {±1} has trivial action gives
O×K/(O×K)2 ' OK×/{±1}(OK×)2× {±1} 'F2[GK].
Corollary 4.1.3. We have Sel2(K)'F2[GK]⊕Cl(K)[2] as F2[GK]-modules.
Proof. SinceF2[GK] is semisimple, the short exact sequence (2.3.1) splits asF2[GK]-modules;
the result then follows from Lemma 4.1.1.
Lemma 4.1.4. For any odd Galois number field K, the GK-invariant subspace of each of the F2[GK]-modules Cl(K)[2], Cl+(K)[2] and Cl4(K)[2] is trivial, whereas the GK-invariant subspace of Cl(K)+4(K) is isomorphic to F2.
Proof. LetC(K) denote one of the groups under consideration, and letC(Q) denote the ray class group of the same modulus but overQ. The norm map induces a group homomorphism C(K)[2] → C(Q)[2], and on GK-invariants it is an isomorphism, with inverse extension of ideals, since n is odd. Indeed, if [a]∈C(K)[2] is GK-invariant, then [Nm(a)] = [a]n = [a]∈ C(K)[2]; similarly, if [(a)] ∈C(Q)[2] then [Nm(aZK)] = [(a)]n = [(a)]∈ C(Q)[2]. Since the groups Cl(Q),Cl+(Q),Cl4(Q) are trivial and Cl+4(Q)'Z/2Z, the result follows.
4.2. First reflection principle. In this section, we show that the Galois module structure of the 2-Selmer group and Kummer duality imply rank inequalities on the class group, classically known as areflection theorem. LetW be an irreducible (left)F2[GK]-module, and for a finitely generated Z[GK]-module M, let rkW(M) ∈Z≥0 be the multiplicity of W in a decomposition of M/2M into irreducible F2[GK]-modules. We recall Lemma 2.1.1, which gives an isomorphism M/2M 'M[2] for a torsion, finitely generated Z(2)[GK]-moduleM, in particular giving rkW(M) = rkW(M[2]).
As mentioned in the introduction, our reflection theorems (Proposition 4.2.2, Proposition 4.3.6, and Theorem 4.3.3) are special cases of the very general T-S-reflection theorem of Gras [23, Th´eor`eme 5.18]. Our goal in the next few sections is to give a direct proof of these results: it shows that they can be read off from the 2-Selmer group, i.e., that they are intrinsic to the underlying structure of the image of the 2-Selmer group, as we will see below.
We use the notation
ρW(K) := rkWCl(K) ρ+W(K) := rkWCl+(K) ρ4,W(K) := rkWCl4(K).
(4.2.1) Proposition 4.2.2. LetK be a Galois number field of odd degree, and letW be an irreducible F2[GK]-module. Then
ρW(K)−ρW∨(K) = rkWF2[GK]−rkW∨S(K) (4.2.3) and
|ρW(K)−ρW∨(K)| ≤rkWF2[GK]. (4.2.4) Proof. Since the short exact sequence in (2.3.1) splits as a sequence of F2[GK]-modules, decomposing Sel2(K) underϕK gives
OK×/(OK×)2 ⊕Cl(K)[2]'Sel2(K)'S(K)⊕ker(ϕK) (4.2.5) as F2[GK]-modules. By Lemma 4.1.1, we have OK×/(OK×)2 ' F2[GK]. By Lemma 3.2.3, we have Cl(K)[2] ' ker(ϕK)∨, so ρW(K) = rkW∨ker(ϕK). Plugging these in and taking rkW∨ in (4.2.5) yields
rkW F2[GK] +ρW∨(K) = rkW∨S(K) +ρW(K) (4.2.6) so
ρW(K)−ρW∨(K) = rkW F2[GK]−rkW∨S(K)≤rkW F2[GK]
giving (4.2.3). Repeating the argument with W∨, noting rkW F2[GK] = rkW∨F2[GK], and
negating then gives (4.2.4).
In particular, we see from the proof of Proposition 4.2.2 that the inequality is refined by the equality4.2.3, with the discrepancy in the inequality being measured by the groupS(K).
This is the simplest instance of the motivation of our paper: we seek to understand structural properties of the 2-Selmer signature map, from which reflection principles are corollaries.
4.3. Isotropy ranks. Similar inequalities govern the narrow class group and its relationship to the class group, encoded in the 2-Selmer group. To measure these contributions, we make the following definitions. Throughout, let W be an irreducible F2[GK]-module.
Definition 4.3.1. The archimedeanW-isotropy rank of K is k+W(K) := rkWCl+(K)−rkW Cl(K) and the 2-adic W-isotropy rank of K is
k4,W(K) := rkW Cl4(K)−rkW Cl(K).
We have k+W(K), k4,W(K)∈Z≥0, since Cl+(K),Cl4(K) surject onto Cl(K).
Proposition 4.3.2. We have
Cl+(K)[2]'Cl(K)[2]⊕(S(K)∩V∞(K))∨, and Cl4(K)[2]'Cl(K)[2]⊕(S(K)∩V2(K))∨. In particular,
k+W(K) = rkW∨(S(K)∩V∞(K)), and k4,W(K) = rkW∨(S(K)∩V2(K)).
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Proof. We have that S(K)∩V∞(K)' ker(ϕK,2)/ker(ϕK); since ker(ϕK,2) and ker(ϕK) are Kummer dual to Cl+(K)[2] and Cl(K)[2] by Lemma 3.2.3, the first isomorphism follows;
taking W-rank and subtracting gives
rkW∨(S(K)∩V∞(K)) = rkW Cl+(K)−rkWCl(K).
The second isomorphism and equality follow similarly.
A further duality is reflected in the totally positive elements in the 2-Selmer group, as follows.
Theorem 4.3.3. Let K be a Galois number field of odd degree. Then Cl+(K)[2]'Cl4(K)[2]∨
as F2[GK]-modules.
Proof. Let Sel+2(K) := ker(ϕK,∞) be the classes in the 2-Selmer group represented by a totally positive element; then Sel+2(K)'Cl4(K)[2]∨by Lemma3.2.3; we show Sel+2(K)'Cl+(K)[2]
as F2[GK]-modules.
Our proof considers the analogue for Sel+2(K) of the exact sequence (2.3.1). Let PK be the group of principal fractional ideals of K, and let PK,>0 be the subgroup ofPK consisting of principal fractional ideals generated by a totally positive element. The map K× → PK sending α 7→ (α) is surjective and GK-equivariant with kernel OK×; it induces the exact sequence
1→ OK×/OK,>0× →K×/K>0× →PK/PK,>0 →1.
By weak approximation, the natural map K×/K>0× → V∞(K) is a GK-equivariant isomor- phism, and so K×/K>0× ' F2[GK] by Proposition 3.1.1(a). Therefore we obtain a isomor- phism of F2[GK]-modules
F2[GK]'(O×K/OK,>0× )⊕(PK/PK,>0). (4.3.4) The natural GK-equivariant map PK →Cl+(K) defined by (α)7→ [(α)] has kernel PK,>0
and so we have a canonical injection PK/PK,>0 ,→Cl+(K). SincePK2 is a subgroup ofPK,>0 the image of the injection is contained in Cl+(K)[2]. Therefore, the map
Sel+2(K)→Cl+(K)[2]/(PK/PK,>0)
mapping the class of α ∈ K× to the class of the fractional ideal a such that a2 = (α) is well-defined; it is also visibly surjective, and so fits into the short exact sequence
1→ O×K,>0/OK×2 →Sel+2(K)→Cl+(K)[2]/(PK/PK,>0)→1 of F2[GK]-modules, giving the isomorphism
PK/PK,>0⊕Sel+2(K)' OK,>0× /O×2K ⊕Cl+(K)[2]. (4.3.5) Adding OK×/O×K,>0 to both sides of (4.3.5), and using (4.3.4) and Lemma4.1.1 we conclude
F2[GK]⊕Sel+2(K)'F2[GK]⊕Cl+(K)[2]
and cancelling gives the result.
By semisimplicity, we can decompose
SK '(SK ∩V∞)⊕(SK∩V2)⊕SK0
as F2[GK]-modules for some choice SK0 ⊆ SK, well-defined up to isomorphism. We call SK0 a coordinate complement to SK in V. With this notation, we immediately turn to our next reflection principle: again, all we use is F2[GK]-module structure and Kummer duality.
Proposition 4.3.6. Let W be an irreducible F2[GK]-module, and let SK0 be a coordinate complement to SK in V. Then
ρ+W(K)−ρ+W∨(K) = rkWF2[GK]−rkW(S∩V∞)−rkW∨(S∩V2)−rkW∨SK0 (4.3.7) and
|ρ+W(K)−ρ+W∨(K)|=|ρ4,W(K)−ρ4,W∨(K)| ≤rkWF2[GK]. (4.3.8) Proof. As K is fixed, we drop it from the notation. By Proposition 4.3.2, we obtain
rkW∨S =k+W +k4,W + rkW∨S0. (4.3.9) From (4.2.6) we get
rkW∨S= rkW F2[GK] +ρW∨ −ρW, so plugging and rearranging gives
ρ+W =k+W +ρW = rkW F2[GK] +ρW∨ −k4,W −rkW∨S0
ρ+W −ρ+W∨ = rkW F2[GK]−k+W∨ −k4,W −rkW∨S0 ≤rkWF2[GK]. (4.3.10) Repeating with W replaced by W∨ gives the inequality |ρ+W(K)−ρ+W∨(K)| ≤ rkWF2[GK] in (4.3.8). By Theorem 4.3.3, we have ρ+W(K) = ρ4,W∨(K) and ρ+W∨(K) = ρ4,W(K), which
gives the equality in (4.3.8) and finishes the proof.
Proposition 4.3.11. Let W be an irreducible F2[GK]-module, and let SK0 be a coordinate complement to SK in V. Then
kW+(K) +kW+∨(K) = rkWF2[GK]−rkW∨SK0 . (4.3.12) Moreover, SK0 is self-dual and
0≤k+W(K) +k+W∨(K) =k4,W(K) +k4,W∨(K)≤rkWF2[GK]. (4.3.13) Proof. We again drop K from the notation. For the equality in (4.3.13), by Theorem 4.3.3, we have
ρ+W +ρ+W∨ =ρ4,W +ρ4,W∨;
subtracting ρW +ρW∨ from both sides gives the result. From (4.3.7) we have k+W +k4,W +ρW −ρW∨ = rkW F2[GK]−rkW∨S0
But k4,W +ρW =ρ4,W =ρ+W∨ by Theorem4.3.3, so
kW+ +k+W∨ = rkW F2[GK]−rkW∨S0 ≤rkW F2[GK]
giving (4.3.12). To restore symmetry, we repeat the same argument with W∨ and conclude
that rkW S0 = rkW∨S0, so in fact S0 is self-dual.
Just as in Proposition 4.2.2, we see from the proof of Proposition (4.3.11) that the real content lies in the equality (4.3.12), i.e., the discrepancy in the upper bound (4.3.13) is measured by the (noncanonically defined) “diagonal subspace” S0(K)⊆S(K).
We deduce corollaries of these statements in the abelian case in section 5.4.
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